Title: #
Understanding Ribbon Categories with 🎀
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In category theory and abstract algebra in general, it can sometimes be very difficult to know when a composition of functions can be simplified. Then even if we know that this composition does simplify, writing a proof for the equality might be a notational nightmare which results in a huge string of function compositions which can be extremely difficult to understand. Fortunately, there is a way to depict these function compositions using ribbons when the category you are working with has enough structure. These types of categories are called strict ribbon categories for this reason. When an isotopy exists between two ribbon diagrams, those two diagrams correspond to an equality of functions in the category. Therefore, we can use these ribbon diagrams to better understand long compositions of functions and provide understandable proofs for why certain compositions can be simplified. In this talk we will establish the relationship between strict ribbon categories and ribbon diagrams by looking at the definition of a ribbon category and associating to each requirement a property of ribbon diagrams. This talk will follow the definition of a strict ribbon category as defined by Turaev in his book, Quantum Invariants of Knots and 3-Manifold. The talk is designed for undergraduate students who have seen at least linear algebra and a proof based course.